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 Introduction to Analysis-of-Variance Procedures

## General Linear Models

An analysis-of-variance model can be written as a linear model, which is an equation that predicts the response as a linear function of parameters and design variables. In general,

where yi is the response for the ith observation, are unknown parameters to be estimated, and xij are design variables. Design variables for analysis of variance are indicator variables; that is, they are always either 0 or 1.

The simplest model is to fit a single mean to all observations. In this case there is only one parameter, , and one design variable, x0i, which always has the value of 1:

The least-squares estimator of is the mean of the yi. This simple model underlies all more complex models, and all larger models are compared to this simple mean model. In writing the parameterization of a linear model, is usually referred to as the intercept.

A one-way model is written by introducing an indicator variable for each level of the classification variable. Suppose that a variable A has four levels, with two observations per level. The indicator variables are created as follows:

 Intercept A1 A2 A3 A4 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1

The linear model for this example is

To construct crossed and nested effects, you can simply multiply out all combinations of the main-effect columns. This is described in detail in "Specification of Effects" in Chapter 30, "The GLM Procedure."

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